Asymptotic results for certain first-passage times and areas of renewal processes

Authors:
Claudio Macci and Barbara Pacchiarotti

Journal:
Theor. Probability and Math. Statist. **108** (2023), 127-148

MSC (2020):
Primary 60F10, 60F05, 60K05

DOI:
https://doi.org/10.1090/tpms/1189

Published electronically:
May 2, 2023

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Abstract: We consider the process $\{x-N(t):t\geq 0\}$, where $x\in \mathbb {R}_+$ and $\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\tau (x),A(x))$ where $\tau (x)$ is the first-passage time of $\{x-N(t):t\geq 0\}$ to reach zero or a negative value, and $A(x)≔\int _0^{\tau (x)}(x-N(t))dt$ is the corresponding first-passage (positive) area swept out by the process $\{x-N(t):t\geq 0\}$. We remark that we can define the sequence $\{(\tau (n),A(n)):n\geq 1\}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $x\to \infty$ in the fashion of large (and moderate) deviations.

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Additional Information

**Claudio Macci**

Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy

Email:
macci@mat.uniroma2.it

**Barbara Pacchiarotti**

Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy

Email:
pacchiar@mat.uniroma2.it

Keywords:
Large deviations,
moderate deviations,
joint distribution,
integrated random walk

Received by editor(s):
October 2, 2021

Accepted for publication:
April 5, 2022

Published electronically:
May 2, 2023

Additional Notes:
The authors acknowledge the support of GNAMPA-INdAM and of MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006).

Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv